Analytical Solution for 2D and 3D Lamb Problem in Saturated Soil Incorporating Effects of Compressibility of Solid and Pore Fluid

Abstract

In this paper, to avoid the complexity, a simple and efficient analytical solution is derived for both 2D and 3D Lamb problems, respectively, in saturated soil under harmonic excitations. Unlike the existing solutions, the proposed solutions for both 2D and 3D Lamb problems in saturated soil under harmonic excitation are capable of well revealing the effect of compressibility of both liquid phase and solid phase on the ground displacements. By applying Fourier transforms and Hankel transforms on the governing equations of wave propagation in saturated soil, wave equations are transformed to ordinary differential equations. Combining the boundary conditions and draining conditions on the ground surface, the displacement solutions on the surface of saturated porous soil due to line and point harmonic excitations are derived, respectively. Then, the solutions in frequency domain are obtained by inverse integral transforms. In the meanwhile, for the sake of discussion without losing its generality, the nondimensional solutions for three-dimensional Lamb problem are derived. The effectiveness and accuracy of the proposed solutions are demonstrated by employing three different approaches. Finally, parametric studies are conducted to investigate the effects of the governing parameters (i.e., exciting frequency, bulk modulus of soil matrix, and bulk modulus of pore fluid) on variation of non-dimensional displacement with the increasing distance away from the excitation source. The results indicate that, in contrast to the effect of the compressibility of soil matrix, the exciting frequency as well as the compressibility of the pore fluid play significant role in affecting the variation of displacement on ground surface subjected to excitations, which particularly highlights that the compressibility of the pore fluid should be carefully considered for evaluating the ground movements.